All fluids are compressible and when subjected to a pressure field causing them to flow, the fluid will expand or be compressed to some degree. The acceleration of fluid elements in a given pressure gradient is a function of the fluid density, ρ, whereas the degree of compression is determined by the isentropic bulk modulus of compression, κ. The speed of sound in a medium is given by, a = (κ/ρ)^{1/2} and compressibility effects are apparent when the flow velocity, u, becomes significant compared to the local speed of sound. The local Mach number M = u/a is the primary parameter which characterizes the effects of compressibility. Under normal atmospheric conditions, the speed of sound in water is 1500 ms^{−1} and that in air is 345 ms^{−1}. Thus, it can be expected that compressibility manifests itself in gas flows more readily than in liquid flows and the discussion below deals predominantly with gas flows. Transients in hydraulic systems are an example of compressible liquid flow which is of some importance. The case of liquid-gas mixtures is of interest and is discussed below.
The role of Mach number in compressible gas flow may be derived from the governing equations of motion and state. However, the physics of these processes are clear when gas flow from one chamber to another is considered. Flow from a constant pressure reservoir, a, is produced by reducing the pressure in chamber b below that in a (Figure 1).
An element of gas, c, will accelerate from a to b, and while doing so increases its volume and decreases temperature. The local velocity of sound reduces as a result of this fall in temperature as a = (γRT)^{1/2}, where γ is the ratio of specific heats, T is the absolute temperature and R is the specific gas constant. Initially, at low-pressure differentials, the flow is essentially incompressible and pressure falls as the gas passes through the throat, d. But it recovers somewhat upon diffusion in the nozzle into chamber b. When the pressure in b is reduced further, this fact is conveyed by sound waves which travel back through the nozzle into a. The flowfield then responds by passing more gas through the nozzle. This process will continue as the pressure in b is reduced up to a point when the local speed of sound has fallen to a level equal to the local flow velocity. This will first occur at the throat, d, and henceforth, the flowfield upstream of the throat will remain frozen as sound waves with the information about conditions in b cannot travel through the throat against the flow. From then on, mass flow through the nozzle remains constant and the nozzle is said to be choked. Reducing the pressure in b further does not increase the mass flow. The flow at the throat has a unity Mach number, i.e., a = u; upstream the flow is subsonic, M < 1, and downstream the flow becomes supersonic, M > 1. (See also Nozzles.)
The same conclusion may be drawn from a one-dimensional isentropic analysis of the steady nozzle flow in Figure 1. This leads to the expression below for du, the change in velocity, with u, resulting from a change, dA, in the nozzle cross-sectional area, A.
At low Mach numbers, M << 1, u varies inversely with A as demanded by incompressible continuity. For subsonic conditions, M < 1, u increases with decreasing A and vice versa. However, the reverse is true for the supersonic case when M > 1. Also, it is only possible for U to keep increasing through the throat, dA = 0, if M = 1; otherwise du must be zero. The conclusion is therefore the same as that arrived at above; the nozzle is choked and the Mach number is unity at the minimum area if the pressure ratio exceeds a particular value.
The isentropic nature of the flow is based on the assumption that no heat transfer between elements of the gas occurs and that the expansion is reversible, i.e., the normal conditions for an isentropic change. Thus, no heat conduction or viscous effects occur. This is true in the free-stream remote from the walls and boundary layer. However, when shock waves or low-density effects are present, this will not be the case.
In any flowfield, a narrow stream tube may be taken such that conditions may be considered one-dimensional (see Figure 1). The Steady Flow Energy Equation (i.e., a control volume for energy) may be applied from the stagnation conditions at the inlet (suffix 0) to any point along the stream tube to give the result:
where h is the specific enthalpy of the fluid. For a perfect gas, h = γRT/(γ − l). Hence, the variation of local static temperature, T, throughout the flowfield—in terms of the local stagnation temperature and local Mach number—immediately follows.
The assumption of isentropic flow then gives the local pressure, P, and density, ρ, in terms of the stagnation values of these quantities. Thus, the ratio of stagnation pressure to local pressures.
Values calculated from these relationships are tabulated in many texts. Anderson (1990), Shapiro (1954), Liepmann and Roshko (1960) have provided such tables. As mentioned in the physical description of nozzle flow, information about boundary conditions is transmitted by sound waves travelling through the flow to adapt the flowfield. In supersonic flow, these waves will always be swept downstream. If a body is introduced into supersonic flow, there has to be a mechanism whereby the upstream flow becomes aware of the presence of the body. Shock waves, being discontinuities in flow parameters, provide this mechanism. In the case of slender and pointed bodies, the shock wave may attach to the leading edge whereas in blunt bodies, the shock wave detaches and stands upstream as a normal shock (Figure 2).
The flow passing through a normal shock is subject to large gradients in temperature and the assumption of isentropic flow is not tenable. The energy conservation equation, together with the mass continuity and momentum equations, and the equation of state for an ideal gas lead to relationships between like properties on either side of the shock; u_{2}/u_{1}, T_{2}/T_{1}, P_{2}/P_{1} (suffices 1 and 2 refer to static conditions upstream and downstream). These may be given in terms of M_{s} = u_{1}/a_{1}, the flow Mach number relative to the stationary shock wave, as the independent variable. Tables of these quantities are given in all standard texts. For a normal shock, the downstream flow is always subsonic in shock relative coordinates.
When supersonic flow expands around a surface convex to the flow (Figure 3), the information is again transmitted along the sound waves, called Mach waves, which travel into the flow.
Conditions along a streamline obey the isentropic equation mentioned previously. However, there is now a direct relationship between the angle the flow has turned through and the local flow Mach number. This is the Prandtl-Meyer relationship, which gives the Prandtl-Meyer angle, ν(M), as a function of the Mach number. The angle ν is the angle the flow has turned through from sonic conditions (M = 1) to the local Mach number. Thus, if flow at M_{1} turns through a further angle, θ, the change in Prandtl-Meyer angle equals q and this enables the new Mach number M_{2} to be determined
The Mach waves travel into the flow causing the turning as shown in Figure 3 and have an inclination to the local flow, μ, called the Mach angle, given by μ = sin^{−1}(l/M).
When two such convex surfaces are arranged to constitute a nozzle, the Mach waves can be seen to interact. This is shown in Figure 4 where it is clear that the streamline on the center line has been subjected to Mach waves from both surfaces, which give no net turning. Nevertheless, the local Mach number will increase according to the sum of the modulus of the turning angles from both walls for there is no physical difference in the flow being turned and expanded by the upper or lower wall.
The process described can be used to design nozzle shapes by calculating the trajectory of the Mach waves and the resulting turning of the flow and is a simple example of the Method of Characteristics for solving supersonic flowfields.
In two dimensions, the shock waves may be inclined and determination of the conditions behind the shock wave is achieved by considering those components of velocity along and perpendicular to the shock separately. The momentum along the shock is unaltered whereas the normal component may be considered as in a normal shock. The results are often given graphically as β, the angle of the shock to the flow as a function of the flow deflection angle, θ. The upstream Mach number. M_{1}, is the independent variable. Such a plot is given in Figure 5.
It can be seen that there are two solutions of β for each value of θ at a given free-stream Mach number, M_{1}. For attached shocks, it is usually the lower value of β—the weak shock—which is relevant. Flow is predominantly supersonic behind the weak shock. When the flow deflection angle is increased above a certain angle, there is no solution. Thus, for sharp bodies of large angle, the shock wave cannot be attached and a normal shock is formed (Figure 6).
Inclined shocks may reflect off solid boundaries in a "regular" manner, which satisfies the condition that the flow remains in contact with the wall after reflection. As there is a restriction on the maximum value of θ, as shown in Figure 5, some regular reflections are not possible and a normal shock forms which is normal to the wall. This automatically satisfies the requirement that the flow remains in contact with the wall. The normal shock then leaves the wall and curves to join the incident shock as shown in Figure 6. These shock formations have their counterparts in supersonic jet plumes. Here, there may be regular reflections from the centerline or a normal shock, called a Mach disk in the axisymmetric situation, could be present. These are also shown in Figure 6.
Isentropic flow which has constant stagnation enthalpy can be shown to be irrotational. This results from Crocco's theorem, and can apply to a wide range of compressible flows, such as nozzle flow. The irrotational nature of the flow means that the velocity may be derived by taking the gradient of a potential function (i.e., a scalar function of position). Thus, the governing equations of motion—the Euler equations—may be represented by a differential equation for the potential function, commonly called the velocity potential equation. In subsonic flow, the equation is elliptical whereas in supersonic flow, the equation is hyperbolic. In the latter case, it can be shown that along Mach lines certain quantities are invariant. In Figure 4, two Mach lines or characteristics arise from any point in the flowfield. These will be at angles θ + μ and θ − μ, where θ is the flow angle and μ, the Mach angle. The quantities
are constant along the −ve and +ve characteristics, respectively. Given an initial upstream boundary condition for the flow, it is possible to march downstream, taking into account constraining walls, to solve the flowfield. In the past, this method of characteristics has been solved graphically but is solved numerically at present. The method of characteristics may also be employed in axisymmetric and three-dimensional flowfields. The characteristics are the mathematical counterparts of the sound waves discussed earlier.
The velocity potential equation may be linearized to solve problems where deviations in a uniform stream are small. This applies to thin airfoils in a uniform stream. In this case, it can be shown that with a simple transformation of geometry the linearized equation is the same as that for incompressible flow, i.e., the Laplace equation.
As a result, the pressure coefficient, C_{p}, for Mach number, M, is related to the incompressible value, C_{pO}, by
This is known as the Prandtl-Glauert Rule. In supersonic flow, the pressure coefficient from linearized theory gives the airfoil surface pressure coefficient as a function of deflection angle, α, of the airfoil surface
Transonic flow occurs beyond the point when the airfoil, for example, becomes critical—i.e., when sonic conditions appear on the airfoil. Shock waves can then exist in the supersonic patch which occurs. This usually results in separation of the airfoil boundary layer, with a consequent increase in drag (Figure 7). In the transonic region, the Mach number is extremely sensitive to small changes in flow area.
In hypersonic flow, the free-stream velocity is much greater than the local velocity of sound. This may roughly occur at Mach numbers greater than 5. When expanding from fixed stagnation conditions, the flow velocity tends to the constant value where a_{O} is the stagnation velocity of sound. In the hypersonic limit, the density ratio across a normal shock approaches a constant and the angle of an oblique shock is linearly related to the deflection angle. Compressible flows are described in standard texts. Anderson (1990), Anderson (1989), Liepmann and Roshko (1957) and Shapiro (1954).
Compressible effects are also very important in long ducts subjected to large pressure ratios. Choking of the ducts may occur if this is of sufficient length, as is often the case. Here, viscous effects are important and in the analysis it is usual to assume that fully-developed viscous flow is present. If the flow is adiabatic and the viscous effects are characterized by a constant friction factor, f, this leads to the Fanno flow solution. The salient features of this flow are.
For a given subsonic inlet Mach number to the duct, there is a maximum length of the duct, L_{max}, in nondimensional terms as given by f L/D max, where D is the duct diameter at which the flow becomes sonic, i.e., it is choked.
Similarly, if supersonic flow enters the duct, the flow decelerates and there is also a nondimensional length at which the flow chokes.
Conditions in the duct relative to the choked conditions for different inlet Mach numbers for Fanno flow are tabulated in most texts. These are widely-used for the design of flow in pipework. Even if choked exit conditions do not prevail, the compressible effects may be deduced from these tables. Another extreme is Rayleigh flow, which is frictionless but with heat addition. Shapiro (1954) has covered a wide range of such flows.
Two-phase bubbly flow can have interesting properties since it is highly compressible, due to the gaseous component, and a high density, due to the liquid present. Thus, the velocity of sound may be much lower than the gas or the liquid in isolation. The subject is covered by Van Wijngaarden (1972) and Drew (1983). The flow does act as a compressible fluid of low speed of sound and exhibits shock structure. This velocity may be as low as 20 ms^{−1} in bubbly water; however, the bubbles may slip with respect to the liquid.
A wide range of unsteady compressible flow phenonema exist and these are covered in the works of Glass and Sislian (1994) and Kentfield (1993). (See also Shock tubes.)
REFERENCES
Anderson, J. D. Jr. (1989) Hypersonic and High Temperature Gas Dynamics, McGraw Hill New York.
Anderson, J. D. Jr (1990) Modem Compressible Flow, McGraw Hill New York.
Drew, D. A. (1983) Mathematical Modeling of Two-Phase Flow, Ann. Rev. Fluid Mech. 15, 261-91.
Glass, I. I. and Sistian, J. (1994) Non Stationary Flows and Shock Waves, Oxford Engineering Series 39, Oxford University Press.
Kentfield, J. A. C. (1993) Nonsteady, One-Dimensional Internal, Compressible Flows, Oxford Engineering Science Series, 31, Oxford University Press. New York.
Liepmann, H. W. and Roshko. A. (1956) Elements of Gasdynamics, Galcit Aeronautical Series, Wiley, New York.
Shapiro, A. H. (1954) Compressible Fluid Flow, Volumes I and II, Ronald Press, New York.
Wijngaarden, L. V. (1972) One-Dimensinal Flow of Liquids Containing Small Gas Bubbles, Ann. Rev Fluid Mech 4, 369.
Использованная литература
- Anderson, J. D. Jr. (1989) Hypersonic and High Temperature Gas Dynamics, McGraw Hill New York.
- Anderson, J. D. Jr (1990) Modem Compressible Flow, McGraw Hill New York.
- Drew, D. A. (1983) Mathematical Modeling of Two-Phase Flow, Ann. Rev. Fluid Mech. 15, 261-91.
- Glass, I. I. and Sistian, J. (1994) Non Stationary Flows and Shock Waves, Oxford Engineering Series 39, Oxford University Press.
- Kentfield, J. A. C. (1993) Nonsteady, One-Dimensional Internal, Compressible Flows, Oxford Engineering Science Series, 31, Oxford University Press. New York.
- Liepmann, H. W. and Roshko. A. (1956) Elements of Gasdynamics, Galcit Aeronautical Series, Wiley, New York.
- Shapiro, A. H. (1954) Compressible Fluid Flow, Volumes I and II, Ronald Press, New York.
- Wijngaarden, L. V. (1972) One-Dimensinal Flow of Liquids Containing Small Gas Bubbles, Ann. Rev Fluid Mech 4, 369. DOI: annurev.fl.04.010172.002101